Would it? I guess maybe. I thought having the adamantium injected would kill him, but once it's there... it's fine.
But you're right. He'd have no marrow or anything.
They don't disappear, his powers, it seems they gradually go. The claws are part of his powers.... wonder what happens to them, if anything.
Hmm... and he's lived so long, maybe he wants to die. Wolverine suicidal? Yeah, maybe.